A class period on -smart 3-vectors with familiar approximates Matt Wentzel-Long1, a) and P. Fraundorf1, b) Physics & Astronomy/Center for Nanoscience, U. Missouri-StL (63121), St. Louis, MO, USA (Dated: 3 July 2019) Introductory physics students have Newton’s laws drilled into their minds, but historically questions related to relativistic motion and accelerated frames have been avoided. With help from the metric equation, these 3- vector laws can be extended into the relativistic regime as long as one sticks with only one reference frame (to define position plus simultaneity), and considers something students are already quite familiar with, namely: motion using map-frame yardsticks as a function of on of the moving-object. The question here is: How may one class period in an intro-physics class, e.g. as a preview before kinematics or later during a on relativity-related material, be used to put the material we teach into a spacetime-smart context?

CONTENTS

I. introduction 1

II. college physics table 1

III. introphysics in general 2

IV. cautions 3

Acknowledgments 4

A. possible course notes 4 1. spacetime version of Pythagoras’ theorem 4 2. traveler-point kinematics 4 FIG. 1. Accelerometer data from a phone dropped and caught 3. dynamics in flat spacetime 5 three . During the free-fall segments, the accelerometer 4. dynamics in curved spacetime 5 reading drops to zero because gravity is a geometric force (like centrifugal) which acts on every ounce of the phone’s structure, and is hence not detected. The positive spikes occur I. INTRODUCTION when the fall is arrested as the falling phone is caught (also by hand) before it hit the floor. Consider driving a car: When you look at your speedometer, what are you seeing? The reported speed is not relative to some inertial frame with synchronized through the floor. It also fails to detect inertial forces, clocks on the side of the road, since speedometers use like those which push you back into the seat (or to the the rotation of the wheels1 (which make static contact outside of a curve) when your car accelerates (or follows with the road) per unit time on the car’s on-board clocks. a curved path). This is good news, coming from general This “proper” ratio2 of map distance ∆x to traveler time relativity, which says that our accelerometer only detects ∆τ at any speed (e.g. even if lightspeed as for Mr. proper forces but that the “undetected” class of geomet- Tompkins3 was only ' 2.5 mph) turns out to be propor- ric forces (associated with accelerated frames or curved tional to 3-vector momentum ~p, to have no upper limit, spacetime) can in general be approximated locally as if and to also be most simply related to kinetic energy and they are one (or more) proper forces. This honored tradi- the time available for driver and pedestrian to react after tion, of treating geometric forces as proper, was of course the danger becomes apparent. It reduces to map distance started in the 17th by none other than Issac New- ∆x per unit map time ∆t only at low speeds. ton himself. Another remarkable everyday example of the “traveler point” approach is the fact that your phone accelerom- eter cannot detect gravity, as shown in Fig. 1. It only II. COLLEGE PHYSICS TABLE detects the normal force which prevents us from falling For example, in our college algebra-based “basic physics II” class, Walker’s text4 has a chapter near the end on relativity. Our strategy is to introduce the book’s a)[email protected] tools, along with these traveler-point tools, for dealing b)[email protected] with problems of , unidirectional velocity 2

TABLE I. Newton at any unidirectional speed in (1+1)D, from (c∆τ)2 = (c∆t)2 − (∆x)2. Quantity↓ Variable→ standard offering traveler-point version low-speed version time dilation γ ≡ ∆t/∆τ γ = 1/p1 − (v/c)2 γ = p1 + (w/c)2 γ ' 1 (vab+vbc) relative velocities v ≡ ∆x/∆t, w ≡ ∆x/∆τ vac = 2 wac = γacγbc(vab + vbc) vac ' vab + vbc (1+vabvbc/c ) momentum p = mγv p = mw p ' mv total energy E = γmc2 E = γmc2 E ' mc2 2 2 1 2 kinetic energy K = (γ − 1)mc K = (γ − 1)mc K ' 2 mv addition, and relativistic energies/momenta. Length con- traction is off the table, because it requires two extended frames with synchronized clocks. Proper-velocity ~w = γ~v and rest-mass m is used instead of “relativistic mass” to preserve the standard relationship between momentum and velocity, and students are only being asked to mas- ter that subset of problems posed in the book which can solved with or without these “hybrid kinematic” tools, as shown Table I.

III. INTROPHYSICS IN GENERAL FIG. 2. Two 6.5 TeV LHC protons send messages to each other, while passing at proper velocities of about ' 6929 More generally we suggest initial mention (even if only lightyears/traveler-, for a collider energy advantage of in passing) of the “traveler-point variables” (chosen be- Krel/K ' 13, 859 times the energy of stationary target colli- cause they either have frame-invariant magnitudes or be- sion. cause they don’t require synchronized clocks), namely traveler or τ, proper velocity defined as map distance per unit traveler time ~w ≡ ∆~x/∆τ, and the net proper force Σξ~ = m~α felt by on-board accelerometers. These are approximated at low speeds by the more fa- miliar map time t, coordinate velocity ~v ≡ ∆~x/∆t, and net map-based force Σf~ ' ∆~p/∆t. By sticking with displacements ∆~x and simultaneity defined by a single bookkeeper or map reference frame (i.e. the metric), as shown in Table II we can simply describe time-dilation γ ≡ ∆t/∆τ and constant unidirectional proper accelera- tion ~α at any speed, even when there’s no time to explore 3-vector proper velocity/acceleration or multi-frame phe- nomena like length contraction. The unidirectional proper-velocity addition equation given in Table II, for example, allows students to see the advantage of colliders over accelerators in more vis- ceral terms, which may even fire up the imagination of NASCAR fans (think of land and relative speed records for particles) as depicted in the relative velocity illustra- tion of Figure 2 (inspired by an XKCD cartoon). Simi- FIG. 3. Round trip times and a sample thrust profile for a larly the unidirectional equations of constant proper ac- spaceship capable of constant 1-gee acceleration and avoiding celeration given in Table II allow students to easily cal- collision with atoms. culate the map and traveler times elapsed on constant proper-acceleration round trips between stars, as illus- trated in Figure 3. is K = (γ−1)mc2. Remarkably, however, in curved In passing, we should also mention the curious relation- time and in accelerated frames, relations like this also ex- ship between various energies and the time-dilation or press potential energies associated with geometric forces. differential-aging factor γ ≡ ∆t/∆τ. The basic relation- This is easiest to see when standing in an artificial (cen- ship, given in Table II, allows us to say that kinetic energy trifugal) gravity well, where the rotational kinetic energy of motion with respect to inertial frames in flat spacetime from a fixed external point of view looks like a potential 3

TABLE II. Traveler-point dynamics in flat (1+1)D spacetime: Conserved quantities energy E = γmc2 and momentum ~p = m ~w, where differential-aging factor γ ≡ δt/δτ, proper velocity ~w ≡ δ~r/δτ ≡ γ~v, coordinate acceleration ~a ≡ δ~v/δt. In the first 4 rows, τ is traveler time from “rest” with respect to the map frame, and α is a fixed space-like proper acceleration vector. Asterisk means that the (1+1)D relation also works in (3+1)D. relation w  c (1+1)D c ατ map time elapsed t t ' τ t = α sinh[ c ] 1 2 c2 ατ map displacement ~x ~x ' ~vot + 2~at x = α (cosh[ c ] − 1 ) 1 2 ατ aging factor γ ≡ δt/δτ γ ' 1 + 2 (v/c) γ = cosh[ c ] ατ proper velocity ~w ≡ δ~x/δτ ~w ' ~v ' ~vo + ~at w = c sinh[ c ] *momentum ~p ~p ' m~v ~p = m ~w = m(γ~v) 2 1 2 2 *energy E E ' mc + 2 mv E = γmc ~ ~ felt (ξ) ↔ map-based (f) f~ ' ξ~ f~ = ξ~ force conversions *work-energy δE ' Σf~ · δ~x δE = Σf~ · δ~x

*action-reaction f~AB = −f~BA f~AB = −f~BA *map-force:momentum Σf~ = δ~p/δt ' m~a Σf~ = δ~p/δt ;Σξ~ = m~α *felt force:acceleration

1 2 2 energy well of depth U ' 2 mω r to the rotating inhabi- usefulness in the case when there are “oppositely tant. However, it also turns out to be true in a spaceship charged” force-carriers is behind the 19th cen- of length L undergoing constant proper acceleration α, tury distinction between magnetic and electrostatic where the energy to climb from trailing to leading end is fields. 2 ∆U = (∆tleading/∆ttrailing − 1)mc ' mαL, and in the gravity of a non-spinning sphere of mass M and radius • 4th caution: The simultaneity of separately located R, where the escape energy for mass m on the surface events is also frame dependent, i.e. differently mov- (when R is much more than the Schwarzschild radius) is ing observers may disagree on which of two “space- 2 like separated” events came first, just as the filial Wesc = (∆tfar/∆τ −1)mc ' GMm/R, where tfar is time elapsed on the clocks of distance observers. This and the ordering of non-descendant relatives in a family tree 5 kinetic differential-aging factors must, for example, both may disagree on an individual’s generation . be considered when calculating your global-positioning- • 5th caution: Relative 3-vector proper-velocity ad- system location. dition (e.g. between co-moving reference frames) is possible, but may be complicated by both “ changes” which affect component magnitudes, and IV. CAUTIONS by changes in the reference metric (which affect both component magnitudes and directions). Cautions for “traveler-point dynamicists”, especially when considering the vantage point of more than one • 6th caution: If energy is not conserved in an in- “map-frame” or bookkeeper metric: teraction between objects traveling at high speeds, momentum may not be either since differently- • 1st caution: Specify “which clock” when talking moving frames allow trades between energy E and about time elapsed, and which “map frame” when momenta ~p (as well as between motion-through- talking about position. time δt/δτ and motion-through-space δ~x/δτ) be- cause only a sum of both, e.g. from the flat-space • 2nd caution: Try to stick with a single map frame of metric c2 = c2(E/mc2)2 − (p/m)2, is frame invari- yardsticks and bookkeeper or “metric time” clocks. ant. This takes discussion of length contraction and Lorentz transforms (both requiring two extended • 7th caution: Geometric forces like gravity and cen- frames) off the table, but allows 3-vector dynamics trifugal in general only work locally, i.e. in regions to be added. within which your reference spacetime metric is “lo- cally flat”. Extensions are possible, e.g. with tidal • 3rd caution: Like rates of energy change at any and Coriolis forces, by combining forces from sep- speed, map-based forces (magnitude & direction) arate regions. differ from one frame to the next at high speeds, even if the frames are only moving at a constant To provide space for discussing sample problems, speed with respect to one another. This frame and for the development of on-line calculators and dependence actually gives rise to a kinetic ver- simulators to further empower students and intro- sus static breakdown of all proper forces, whose ductory teachers with this metric-first6 or ‘one-map 4 two-clock’7 approach, we’ve created some space up at sites.google.com/umsl.edu/travelerpointdynamics2 for further discussion.

ACKNOWLEDGMENTS

Thanks to the late Bill Shurcliff (1909-2006) for his counsel on “minimally-variant” approaches.

1V. N. Matvejev, O. V. Matvejev, and O. Gron, “A relativistic trolley paradox,” Amer. J. Phys. 84, 419 (2016). 2W. A. Shurcliff, “Special relativity: The central ideas,” (1996), 19 Appleton St, Cambridge MA 02138. 3G. Gamow, Mr. Tompkins in paperback (Cambridge University Press, 1996) illustrated by the author and John Hookham. 4J. S. Walker, Physics, 5th ed. (Addison-Wesley, NY, 2017). 5C. Rovelli, The order of time (Allen Lane, London, 2018). 6E. Taylor and J. A. Wheeler, Exploring black holes, 1st ed. (Ad- dison Wesley Longman, 2001). 7D. G. Messerschmitt, “Relativistic timekeeping, motion, and grav- ity in distributed systems,” Proceedings of the IEEE 105, 1511– FIG. 4. Pythagorean frame-invariance of the hypotenuse 1573 (2017). (in black), from the metric equation (δh)2 = (δx)2 + (δy)2 for seven different Cartesian coordinate systems, whose unit- vectors are shown in color. Although none of these coor- Appendix A: possible course notes dinate systems agrees on the coordinates of the black line’s endpoints, all agree that the length of the hypotenuse is 5. The sections to follow might be dropped into the in- troductory physics schedule as the usual “relativity sec- agree on its even if they can’t agree on which of tion”, near the end of the course, or may be rearranged two spatially-separated events happened first. for piecemeal discussion earlier so that the Newtonian models, that students will be working with, are framed This equation is seriously powerful. As Einstein illus- as approximations from the start. trated, if one curves spacetime by tweaking the “unit” coefficients of the terms on the right by only “one part per billion”, we find ourselves in a gravitational field like that on earth where a fall of only a few meters can do 1. spacetime version of Pythagoras’ theorem you in. Time is local to a given clock, and simultaneity is determined by your choice of reference frame. Al- though Maxwell’s equations on electromagnetism were 2. traveler-point kinematics “informed” to this reality in the mid 1800’s, humans re- ally didn’t start to get the picture until the early 1900’s. From the foregoing, it is easy to define a proper- But how might one deal with this quantitatively? velocity ~w ≡ d~x/dτ = γ~v, where ~v ≡ d~x/dt is Start with the (1+1)D flat-space metric equation, coordinate-velocity as usual, and speed of map- namely (cδτ)2 = (cδt)2 − (δx)2 where x and t are po- time or “differential-aging factor” γ ≡ dt/dτ = p p sition and time coordinates associated with your refer- 1 + (w/c)2 = 1/ 1 − (v/c)2 ≥ 1. This last relation ence “map-frame” of yardsticks and synchronized clocks. tells us that when simultaneity is defined by a network of The quantity τ is the proper-time elapsed on the clocks synchronized map clocks, a moving traveler’s clock will of a traveling observer whose map-position x may be always run slow. These relationships follow directly from written as a function of map-time t. As usual c is the the metric equation itself. spacetime constant (literally the number of meters in a Thus having a new time-variable τ also gives us some ) which is traditionally referred to as lightspeed new ways to measure rate of travel. Proper-velocity, as because it equals the speed of light in a vacuum. we’ll see, has no upper limit and is related to conserved- The on the left in the metric equation is referred quantity momentum by the simple vector relation ~p = to as a frame-invariant. Just like a given hypotenuse m~w where m is our traveler’s frame-invariant rest-mass. (cf. Fig. 4) can be expressed in terms of a bunch of The upper limit of c ≥ dx/dt on coordinate-velocity re- different xy coordinate systems, all of which agree on its sults simply from the fact that momentum (and dx/dτ) length, so a given proper-time interval e.g. on a traveling have to remain finite. Speed of map-time γ, on the other object’s clock, can be expressed in terms of many differ- hand, relates to total energy by E = γmc2, and to ent “bookkeeper” reference-frames, all of which will also kinetic energy by K = (γ − 1)mc2. 5

At this point, a variety of familar time-dilation and 4. dynamics in curved spacetime relativistic energy/momentum topics might be covered as examples. If there is added time, one path to take is to introduce length-contraction, velocity addition, and In both curved spacetime and in accelerated frames, Doppler effect with or without Lorentz transforms, his- Newton’s equations still work (at least locally) provided torical notes on lightspeed measurement, etc. In what that we recognize the existence of non-proper or geomet- follows, we instead push the alignment with traveler-point ric forces, like gravity as well as inertial forces that arise concepts and the Newtonian treatment of kinematics and in accelerated frames. Newton’s law for causes of motion mechanics a bit further. “in the neighborhood of a traveling object” then takes It is also conceptually interesting to note that proper- the 3-vector form ΣF~o + ΣF~g = m~α, where ~α is the net- acceleration, in turn, is simply the vector-acceleration acceleration actually observed by the traveler. detected by a cell-phone accelerometer in the traveler’s The reason that your cell-phone’s accelerometer can’t pocket. This quantity, after a couple of proper-time see gravity (or centrifugal force), even when gravity is derivatives, pops up on the left side of the metric equa- causing a net-acceleration downward, is that these are tion as a frame-invariant as did “hypotenuse” and proper- geometric forces. Geometric forces (ΣF~g), which act on time. “every ounce” of an object, result from being in an ac- For unidirectional motion in flat-spacetime (i.e. us- celerated frame or in curved spacetime. Accelerometers ing map-coordinates in an inertial frame), proper- ~ 3 2 2 can only detect the result of net proper-forces (ΣFo), i.e. acceleration ~αo = γ ~a, where ~a ≡ d ~x/dt is the usual one’s proper-acceleration ~α . coordinate-acceleration. These relations also yield a o few simple integrals for “constant” proper-acceleration, It’s traditional to approximate gravity on the surface of 2 namely α = ∆w/∆t = c∆η/∆τ = c ∆γ/∆x. the earth as simply a proper-force that is proportional to That 2nd equality involves “hyperbolic velocity-” ~ −1 −1 mass m, for which F = m~g where ~g is a downward vector or rapidity η = sinh [w/c] = tanh [v/c], so that with a magnitude of about 9.8 [m/s2]. This works quite γ = cosh[η]. The first and third equalities reduce to the well for most applications. However unlike geometric- familiar conceptual-physics relationships a = ∆v/∆t = forces, proper-forces are not associated with positional 1 ∆(v2)/∆x at low speed. However they allow begin- 2 time-dilation like that which must be figured into GPS ning students to explore interstellar constant proper- system calculations. acceleration round-trip problems, almost as easily as they do problems involving projectile trajectories on earth. Just as kinetic energy in flat spacetime is related via (dt/dτ −1)mc2 to the faster passage of map-time (t) with respect to traveler time (τ) when simultaneity is defined 3. dynamics in flat spacetime by the map-frame, so is the position-dependent potential- energy well-depth of some geometric-forces in accelerat- The net proper-force may in general be written as ing frames and curved spacetime. Thus (dt/dτ − 1)mc2 ΣF~o ≡ m~αo. In flat (and unaccelerated) spacetime also describes the potential-energy well-depth for mass- coordinate-systems, all forces are proper, and proper- m travelers located: (i) at radius r from the axis in a acceleration ~αo equals the net-acceleration ~α observed habitat rotating with angular velocity ω, which in the 1 2 2 by a traveler. Under a constant net proper-force, we can low-speed limit reduces to the classical value 2 mω r ; therefore expect constant net-acceleration. (ii) a distance L behind the leading edge of a spaceship At high speeds the constant proper-force equations are undergoing constant proper acceleration α, which in the messier because of that pesky γ in equations like ∆w = small L limit reduces to the classical value mαL; and ∆(γv) = αo∆t. The low speed approximation (namely (iii) on the radius R surface of a mass M planet, which ∆v = a∆t) is therefore a bit simpler to deal with, and in the R  2GM/c2 limit reduces to the classical value works fine for speeds well below c. of GMm/R where G is the gravitational constant.